Question

A ray of light is incident on a glass prism of refractive index µ and refracting angle A. If it just suffers total internal reflection at the other face, obtain a relation between the angle of incidence, angle of prism and critical angle.

Answer 1

For surface 1:

Snell's law

sin i = µ sin r₁

r₂ = i_c

In quadrilateral OPQR,

∠OPQ + ∠PQR + ∠QRO + ROP = 360°

90° + ∠PQR + 90° + A = 360°

∠PQR = 180° - A

In ΔPQR,

`r_1 + r_2 + 180^circ - A = 180^circ`

`r_1 + r_2 = A`

`r_1 = A - i_c` ....[∵ r₂ = i_c]

sin i = µ sin(A - i_c) .....(i)

For surface 2:

µ sin i_c = sin90°

`mu = 1/sini_c` .....(ii)

`sini = 1/(sini_c) xx sin(A - i_c)` .....[from (i)]

`i = sin^-1((sin(A - i_c))/(sini_c))`